Discrete (h, k)-Dichotomy and Remarks on the Boundedness of the Projections

The present paper treats a concept of -dichotomy for linear discrete systems. Sufficient conditions for the -boundedness of the projection sequences that give the dichotomy are presented and an illustrative example shows the connection between the growth of the system and the bound of the sequence of projections. Thus the growth of the system that is assumed in the theorems is essential.

A natural generalization of both the uniform and nonuniform dichotomy is successfully modeled by the concept of (ℎ, )-dichotomy, where a significant number of papers containing many interesting results were published, from which we mention [5,[13][14][15].In recent years, several papers, also dedicated to asymptotic behavior of discrete dynamical systems, appeared and hence the generalization to (ℎ, )behavior is present even for the trichotomy property (see [3,8,9,[15][16][17]).Among the results that were obtained in this field, we concentrate our attention on the property of boundedness of the sequence of the projections that give the dichotomic behavior.For example, in [10], in the case of discrete variational systems, the uniform exponential dichotomy property of a discrete cocycle {Φ(, )} ∈Θ,∈N which has a uniform exponential growth implies the boundedness of the dichotomic sequence of projections (here, the proof being more direct, compared with those found in the literature, without using the angular distance between the dichotomy subspaces).Further results concerning the boundedness of the dichotomy sequence of the projections are also presented in [4].In [12], such results are proved both in the case of uniformly bounded coefficients of the discrete linear system (see Remark 3.3) and through techniques of admissibility of pairs of function spaces to the associated control system of the discrete linear system (Theorem 3.3).This is the direction in which the present paper intends to state the results, by defining the concept of (ℎ, )-dichotomy for linear discrete systems and showing that, under similar hypotheses as stated in [10], we obtain the -boundedness of the sequence of projections that give the dichotomic behavior.
Bounded and exponentially bounded sequences of projections, although not explicitly stated, are widely used in the study of the exponential dichotomy, and this intrinsic property is assumed from the beginning, in the definition of the asymptotic behavior (see, e.g., [2]).Having in mind the above reasons, we present sufficient conditions of the boundedness of a sequence of projections that give the (ℎ, )dichotomy for a given discrete linear system.Finally, an illustrative example will show that the main assumption of (ℎ, )-growth is essential, and we present an example that, besides the property mentioned above, also shows how closely connected is the growth of the system to the boundedness of the sequence of projections.

Basic Definitions and Notations
Let  be a Banach space and B() the Banach algebra of all bounded linear operators on .The norms on  and on B() will be denoted by ‖ ⋅ ‖.The identity operator on  is denoted by .We also denote by N * the set of positive integers and by R + the set of nonnegative real numbers.We also denote by We consider the linear difference system where A : N → B() is a given sequence.For (, ) ∈ Δ we define Definition 1.The map  : Δ → B() defined above is called the evolution operator associated to the system (A).
Remark 2. If the sequence (  ) ≥0 is a solution of (A), then If  : N → B() is a sequence of projections, then the sequence  : N → B() defined by is also a sequence of projections, called the complementary sequence of .
then we say that  is exponentially bounded.
(ii) If () =  + 1, that is, there exist  > 0 and  ≥ 0 such that then we say that  is polynomially bounded.
(iii) If  = 0, that is, there exists  > 0 such that then we say that  is bounded.
In the particular case in which  2 = 0, we say that the system (A) has a uniform (ℎ, )-growth or an ℎ-growth.
Remark 8.As particular cases of (10), we have the following.
(i) If ℎ() = () =   , then we say that the system (A) has an exponential growth.
(ii) If ℎ() =   , then we say that the system (A) has a nonuniform exponential growth.
(iii) If ℎ() =   and  2 = 0, then we say that the system (A) has a uniform exponential growth.
(v) If ℎ() =  + 1, then we say that the system (A) has a nonuniform polynomial growth.
(vi) If ℎ() =  + 1 and  2 = 0, then we say that the system (A) has a uniform polynomial growth.
As we will see below, the (ℎ, )-dichotomy covers a wide range of dichotomy concepts, which are widely used throughout the papers from the bibliography and the references therein (e.g., [13,14]).
The sufficiency follows from choosing  =  0 .
In what follows we will present the main results of our paper.
If  : N → B() is a family of projections that gives the (ℎ, )-dichotomy of the system (A), then  is -bounded.
If  : N → B() is a family of projections that gives the (ℎ, )-dichotomy of the system (A), then  is -bounded.
Remark 15.In contrast with the condition (c) from Theorem 13, the condition (c) from Corollary 14 is rather restrictive; namely, such a pair (ℎ, ) is quite particular.Indeed, from (c), having in mind that ℎ is a growth rate, it follows that, for all  ∈ N, Let  ∈ N and  =  + 1.It follows that ℎ( + 1)/ℎ() = ℎ(1) from where By setting  = ln ℎ(1) ≥ 0 we obtain that showing us that the behavior of the system (A) is exponential.
Corollary 16.If the system (A) has exponential growth and it is uniformly exponentially dichotomic, then there exists a sequence of projections that gives the specified dichotomic behavior of the system (A) and it is also exponentially bounded.
If  : N → B() is a family of projections that gives the (ℎ, )-dichotomy of the system (A), then  is -bounded.
Proof.Let ,  and  : N → B() be given from the uniform (ℎ, )-dichotomy property and the constants ,  1 ,  2 form the (ℎ, )-growth property of (A).Let  ∈ N and  ∈  be fixed.As in the proof from Theorem 13 we obtain the inequality valid for all  ≥ .
Corollary 18.If the system (A) has polynomial growth and it is uniformly polynomially dichotomic, then there exists a sequence of projections that gives the assumed polynomial behavior of the system (A) which is polynomially bounded.Indeed, if () = ℎ() =  + 1, it is easy to see that and by applying the preceding result, we obtain the desired conclusion.
Remark 19.If, for the system (A), the (ℎ, )-growth is uniform and the (ℎ, )-dichotomy is as well uniform, then the given sequence of projections that gives the dichotomy property is bounded.
Remark 20.In order to prove the -boundedness of the sequence of projections in the case of uniform asymptotic behavior, the (ℎ, )-growth is essential, as we will see in the following example.Moreover, by the following choice of the system to depend on the sequence of projections, we also intend to point out how the growth and the dichotomy are strongly connected to each other, through the same growth rate .
We will give a particular example, in the exponential framework.
Example 21.Consider the growth rates ℎ,  : R + → [1, ∞) defined by ℎ () =   , () = ( + 1) 2  ∈ N. (30) On  = R 2 endowed with the norm define the sequence of projections  : N → B(R 2 ) by Let ,  ∈ N and  = ( 1 ,  2 ) ∈ R 2 .We observe that hence Moreover, if  : N → B(R 2 ) is the complementary sequence of , defined by we have that for all ,  ∈ N hence As a consequence of the above, we also have that for all ,  ∈ N In addition, another property that we will use in what follows is given by the fact that if (, ) ∈ Δ and  = ( Consider the sequence of operators A : N → B() given by and the corresponding linear system (A).Next we will give the expression of the evolution operator which governs (A).
For (, ) ∈ Δ with  ̸ = , one can see that The -boundedness of the sequence of projections  fails to hold, mainly because the (ℎ, )-growth of the system (A) does not hold.